What is multiple importance sample option
Multiple importance sample (MIS) basically helps guide rays towards light sources, reducing noise
MIS only applies if the material has an emission node. (world materials and lamps behave a little differently).
From the wiki:
Material

By default objects with emitting materials use both direct and indirect light sampling methods, but in some cases it may lead to less noise overall to disable direct light sampling for some materials. This can be done by disabling the Multiple Importance Sample option. This is especially useful on large objects that emit little light compared to other light sources.
This option will only have an influence if the material contains an emission node; it will be automatically disabled otherwise.

Here's an example with a bright light source (a sphere) and a weak light source (a plane above the scene, you can see it reflected in the seconds sphere)
Without MIS on the low strength plane:

With MIS on the low strength plane:

The bright sphere has MIS enabled in both examples. The second example seems to have slightly more noise (look at the shadow of the second sphere and the reflection of the floor plane)
World

By default lighting from the world is computed solely with indirect light sampling. However for more complex environment maps this can be too noisy, as sampling the BSDF may not easily find the highlights in the environment map image. By enabling this option, the world background will be sampled as a lamp, with lighter parts automatically given more samples.

Without MIS:

With MIS:

The wiki really says it all here.
Lamp

By default lamps use only direct light sampling. For area lights and sharp glossy reflections, however, this can be noisy, and enabling this option will enable indirect light sampling to be used in addition to reduce noise.

Without MIS only rays that bounce directly to the lamp are sampled (camera &gt; surface &gt; lamp):

With MIS rays that bounce indirectly are sampled too (e.g. camera &gt; surface &gt; surface &gt; lamp):